Sum of 1/n^2

Date: 07/24/2000 at 06:35:53
From: Mike
Subject: Sum of 1/n^2 without Fourier series
Sirs,
Euler proved that the sum of 1/n^2 is equal to pi^2/6. It is easily
shown that this sum is equal to INT(0->1)INT(0->1) 1/(1-xy) dxdy or
the limit of the double integrals over the rectangle [0,t]X[0,t] as
t->1(from the left).
I made the change of variables x=(u-v)/(2)^1/2 and y=(u+v)/(2)^1/2,
which rotates the original rectangle about the origin by pi/4 in the
uv plane. In changing the variables, the integral changed from f(x,y)
to 2/(2-u^2+v^2); the Jacobian is 1, but in changing the bounds, you
go from [0,1]X[0,1] to four functions in uv: v = u, v = -u, v = 1-u
and v = u-1. which I do not know how to interpret in the bounds of a
double integral.
Also, how should I start in evaluating the integral? In its first
integration (say wrt v), it is in the form of 2/(c+v^2), where
c = 2+u^2, which is close to arctan(v) as a solution.
I hope I have explained this clearly.
Thank you for your time.

Date: 07/24/2000 at 11:26:25
From: Doctor Rob
Subject: Re: Sum of 1/n^2 without Fourier series
Thanks for writing to Ask Dr. Math, Mike.
The square over which you are integrating has sides with inclination
Pi/4 and -Pi/4. To integrate over this region, you should probably
split it into two regions as follows:
0 <= u <= sqrt(2)/2, -u <= v <= u,
and
sqrt(2)/2 <= u <= sqrt(2), u - sqrt(2) <= v <= sqrt(2) - u.
Now to do the inside integration with respect to v, write
2/(2+u^2-v^2) = A/(sqrt[2+u^2]+v) + B/(sqrt[2+u^2]-v),
A = -1/sqrt[2+u^2],
B = 1/sqrt[2+u^2].
Then the integral is
A*ln(sqrt[2+u^2]+v) + B*ln(sqrt[2+u^2]-v),
evaluated between the limits. This can be simplified to
4*Arctanh(u/sqrt[2+u^2])/sqrt(2+u^2) +
4*Arctanh([sqrt(2)-u]/sqrt[2+u^2])/sqrt(2+u^2).
Now the first term can be integrated, and you get
2*Arctanh[1/Sqrt(5)]^2, but the second term seems intractable.
A variation of the method you suggest is actually successful in
computing SUM 1/n^2. The trick is to compute the sum of the
reciprocals of the squares of the odd numbers only:
infinity
S = SUM 1/(2*n-1)^2
n=1
This can be shown to be identical to the double integral
1 1
S = INTEGRAL INTEGRAL 1/(1-x^2*y^2) dy dx,
0 0
using the same method. Now there is a very clever way to evaluate
this double integral: use the substitution
x = sin(u)/cos(v),
y = sin(v)/cos(u).
The Jacobian of this transformation is 1 - tan^2(u)*tan^2(v), and the
integrand becomes 1/(1-tan^2[u]*tan^2[v]), so the new integrand is
magically reduced to 1. The only problem is to figure out the new
limits of integration. It turns out that the region over which to
integrate is the triangle bounded by u = 0, v = 0, and u + v = Pi/2.
Integrating 1 over this region gives the area of the triangle, which
is obviously Pi^2/8 (half the base Pi/2 times the height Pi/2). Thus
S = Pi^2/8.
Now to find the actual sum you want, call it T, observe that
T - T/4 = S,
T = 4*S/3 = Pi^2/6.
- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/